Question

Create a vector-valued function whose graph matches the given description. A circle of radius 2, centered at \((1,2),\) traced counterclockwise once on \([0,2 \pi]\).

Short answer

\(\mathbf{r}(t) = \langle 1 + 2 \cos(t), 2 + 2 \sin(t) \rangle\) for \(t \in [0, 2\pi]\).

Step-by-step solution

Understand the Circle Equation

The given problem involves a circle. The general parametric equation for a circle of radius \(r\) centered at \((h, k)\) in terms of angle \(t\) is \((x, y) = (h + r \cos(t), k + r \sin(t))\). This equation helps trace the circle over an interval.

Apply the Calculations

We need to set the circle parameters to match the provided description:- Radius \(r = 2\)- Center \((h, k) = (1, 2)\)- Interval \([0, 2\pi]\).Thus, the parametric equations become:\[x(t) = 1 + 2 \cos(t)y(t) = 2 + 2 \sin(t)\]

Formulate the Vector-Valued Function

To express these parametric equations as a vector-valued function, we combine them into one vector function:\[\mathbf{r}(t) = \langle x(t), y(t) \rangle = \langle 1 + 2 \cos(t), 2 + 2 \sin(t) \rangle\] for \(t \in [0, 2\pi]\).

Verify the Tracing Direction and Interval

By construction, the circle is traced counterclockwise because the parameter \(t\) increases from \(0\) to \(2\pi\). At \(t=0\), the point is \((3, 2)\) (to the right of the center), and at \(t=\frac{\pi}{2}\), it is \( (1, 4)\) (above the center), confirming counterclockwise direction.

Key concepts

Parametric Equations

Parametric equations allow us to describe geometrical shapes in terms of parameters. In the context of a circle, these parameters typically involve trigonometric functions, such as sine and cosine. These functions oscillate between -1 and 1, creating the perfect setup to express circular motions. For a circle with a radius of \(r\) centered at \((h, k)\), the parametric equations are:

• \( x(t) = h + r \cos(t) \)
• \( y(t) = k + r \sin(t) \)

These equations link the angle \(t\), which moves from \(0\) to \(2\pi\), to specific \(x\) and \(y\) coordinates on the circle. This way, as \(t\) sweeps through its range, we trace the circle's perimeter, capturing the essence of circular motion.

Circle Equation

The equation of a circle is a mathematical way to express all points that are equidistant from a center point. When we describe a circle using parametric equations, the aim is to represent its position and size. Given a center \((h, k)\) and a radius \(r\), the circle's parametric equations become:

• \( x(t) = h + r \cos(t) \)
• \( y(t) = k + r \sin(t) \)

The center point \( (h, k) \) defines where the circle is placed in the coordinate plane. The radius \(r\) specifies how far the circle reaches from this center point. In essence, these equations help in constructing a circle in a parametric manner, capturing both location and size efficiently.

Counterclockwise Tracing

When a circle is traced using parametric equations, the direction of tracing depends on how the parameter \(t\) progresses. For counterclockwise tracing, \(t\) moves from \(0\) to \(2\pi\), following the natural order of the trigonometric circle. To verify counterclockwise movement:

• At \(t=0\), the circle reaches its right-most point, directly at \((h + r, k)\).
• At \(t=\frac{\pi}{2}\), it rises to its top point at \((h, k + r)\).

This progression means that as \(t\) increases, it mimics a counterclockwise rotation, sweeping from right to up, then left and downward, concluding a full rotation. Understanding this is crucial for visualizing how a parametric equation moves through the circle's path.

Center of the Circle

The center of a circle in the parametric form is the anchor around which the circle is drawn. Located at the point \((h, k)\), it is a constant in the parametric equations. This means, regardless of the value of the parameter \(t\), the distance from any point on the circle to this center point is always exactly the radius \(r\). Understanding this can simplify how we view circular paths:

• The circle's position in space is determined by this center point.
• The equation \(x(t) = h + r \cos(t)\) provides horizontal offsets from \(h\).
• Similarly, \(y(t) = k + r \sin(t)\) gives vertical offsets from \(k\).

Therefore, determining the center allows for constructing and visualizing the circle's location and orientation on a coordinate plane.